Mail autosync question

Im new to mac and I dont understand everything yet. I want to ask you a question about the Mail-program.
The problem is that when I boot up my MacBook, the program doesnt show me automaticly when I have recieved new e-Mails (on the round red indicator, in the edge of the icon). Not until I finally launch the program, the indicator and the notifications appear.
So, is it possible to see if I have recieved new e-Mails before I have to open the program? Maybe that the mails get automaticly synced without the program being running?!

I see now that it is a common issue on macs.
And sadly there seems to be no perfect solution for Mail-notifications, unles I certainly wont pay 9 for Mail Call in the generally totally overpriced App Store.

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